Is a voluntary circumambulation (tawaf al-tatawwu’) a substitute for the umrah circumambulation (tawaf al-umrah), and what is incumbent upon someone from whom secretions invalidating ablution (wudu’) are emitted during the circumambulation?

Scholars have differed regarding the ruling on purity for circumambulation (Tawaf). The majority of them are of the view that it is a condition for the validity of Tawaf, while some hold that it is only a Sunnah, and that the Tawaf of one who is in a state of minor ritual impurity is valid. This is the choice of Sheikh Al-Islam Ibn Taymiyyah and the preferred opinion of العلامة Ibn Uthaymeen. Based on this, your 'Umrah is valid, and nothing is required of you, and your second Tawaf is supererogatory.